\(\int \frac {\sqrt {1+a x}}{\sqrt {x}} \, dx\) [222]

Optimal result

Integrand size = 15, antiderivative size = 34 \[ \int \frac {\sqrt {1+a x}}{\sqrt {x}} \, dx=\sqrt {x} \sqrt {1+a x}+\frac {\text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}} \]

[Out]

arcsinh(a^(1/2)*x^(1/2))/a^(1/2)+x^(1/2)*(a*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {52, 56, 221} \[ \int \frac {\sqrt {1+a x}}{\sqrt {x}} \, dx=\frac {\text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}+\sqrt {x} \sqrt {a x+1} \]

[In]

Int[Sqrt[1 + a*x]/Sqrt[x],x]

[Out]

Sqrt[x]*Sqrt[1 + a*x] + ArcSinh[Sqrt[a]*Sqrt[x]]/Sqrt[a]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 56

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps \begin{align*} \text {integral}& = \sqrt {x} \sqrt {1+a x}+\frac {1}{2} \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx \\ & = \sqrt {x} \sqrt {1+a x}+\text {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right ) \\ & = \sqrt {x} \sqrt {1+a x}+\frac {\sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.41 \[ \int \frac {\sqrt {1+a x}}{\sqrt {x}} \, dx=\sqrt {x} \sqrt {1+a x}+\frac {2 \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x}}{-1+\sqrt {1+a x}}\right )}{\sqrt {a}} \]

[In]

Integrate[Sqrt[1 + a*x]/Sqrt[x],x]

[Out]

Sqrt[x]*Sqrt[1 + a*x] + (2*ArcTanh[(Sqrt[a]*Sqrt[x])/(-1 + Sqrt[1 + a*x])])/Sqrt[a]

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.21

method	result	size
meijerg	\(-\frac {-2 \sqrt {\pi }\, \sqrt {a}\, \sqrt {x}\, \sqrt {a x +1}-2 \sqrt {\pi }\, \operatorname {arcsinh}\left (\sqrt {a}\, \sqrt {x}\right )}{2 \sqrt {a}\, \sqrt {\pi }}\)	\(41\)
default	\(\sqrt {x}\, \sqrt {a x +1}+\frac {\sqrt {\left (a x +1\right ) x}\, \ln \left (\frac {\frac {1}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+x}\right )}{2 \sqrt {a x +1}\, \sqrt {x}\, \sqrt {a}}\)	\(57\)
risch	\(\sqrt {x}\, \sqrt {a x +1}+\frac {\sqrt {\left (a x +1\right ) x}\, \ln \left (\frac {\frac {1}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+x}\right )}{2 \sqrt {a x +1}\, \sqrt {x}\, \sqrt {a}}\)	\(57\)

[In]

int((a*x+1)^(1/2)/x^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2/a^(1/2)/Pi^(1/2)*(-2*Pi^(1/2)*a^(1/2)*x^(1/2)*(a*x+1)^(1/2)-2*Pi^(1/2)*arcsinh(a^(1/2)*x^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.65 \[ \int \frac {\sqrt {1+a x}}{\sqrt {x}} \, dx=\left [\frac {2 \, \sqrt {a x + 1} a \sqrt {x} + \sqrt {a} \log \left (2 \, a x + 2 \, \sqrt {a x + 1} \sqrt {a} \sqrt {x} + 1\right )}{2 \, a}, \frac {\sqrt {a x + 1} a \sqrt {x} - \sqrt {-a} \arctan \left (\frac {\sqrt {a x + 1} \sqrt {-a}}{a \sqrt {x}}\right )}{a}\right ] \]

[In]

integrate((a*x+1)^(1/2)/x^(1/2),x, algorithm="fricas")

[Out]

[1/2*(2*sqrt(a*x + 1)*a*sqrt(x) + sqrt(a)*log(2*a*x + 2*sqrt(a*x + 1)*sqrt(a)*sqrt(x) + 1))/a, (sqrt(a*x + 1)*
a*sqrt(x) - sqrt(-a)*arctan(sqrt(a*x + 1)*sqrt(-a)/(a*sqrt(x))))/a]

Sympy [A] (verification not implemented)

Time = 0.82 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {1+a x}}{\sqrt {x}} \, dx=\sqrt {x} \sqrt {a x + 1} + \frac {\operatorname {asinh}{\left (\sqrt {a} \sqrt {x} \right )}}{\sqrt {a}} \]

[In]

integrate((a*x+1)**(1/2)/x**(1/2),x)

[Out]

sqrt(x)*sqrt(a*x + 1) + asinh(sqrt(a)*sqrt(x))/sqrt(a)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (24) = 48\).

Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.00 \[ \int \frac {\sqrt {1+a x}}{\sqrt {x}} \, dx=-\frac {\log \left (-\frac {\sqrt {a} - \frac {\sqrt {a x + 1}}{\sqrt {x}}}{\sqrt {a} + \frac {\sqrt {a x + 1}}{\sqrt {x}}}\right )}{2 \, \sqrt {a}} - \frac {\sqrt {a x + 1}}{{\left (a - \frac {a x + 1}{x}\right )} \sqrt {x}} \]

[In]

integrate((a*x+1)^(1/2)/x^(1/2),x, algorithm="maxima")

[Out]

-1/2*log(-(sqrt(a) - sqrt(a*x + 1)/sqrt(x))/(sqrt(a) + sqrt(a*x + 1)/sqrt(x)))/sqrt(a) - sqrt(a*x + 1)/((a - (
a*x + 1)/x)*sqrt(x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (24) = 48\).

Time = 6.28 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.91 \[ \int \frac {\sqrt {1+a x}}{\sqrt {x}} \, dx=-\frac {a {\left (\frac {\log \left ({\left | -\sqrt {a x + 1} \sqrt {a} + \sqrt {{\left (a x + 1\right )} a - a} \right |}\right )}{\sqrt {a}} - \frac {\sqrt {{\left (a x + 1\right )} a - a} \sqrt {a x + 1}}{a}\right )}}{{\left | a \right |}} \]

[In]

integrate((a*x+1)^(1/2)/x^(1/2),x, algorithm="giac")

[Out]

-a*(log(abs(-sqrt(a*x + 1)*sqrt(a) + sqrt((a*x + 1)*a - a)))/sqrt(a) - sqrt((a*x + 1)*a - a)*sqrt(a*x + 1)/a)/
abs(a)

Mupad [B] (verification not implemented)

Time = 12.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {1+a x}}{\sqrt {x}} \, dx=\sqrt {x}\,\sqrt {a\,x+1}+\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {a\,x+1}-1}\right )}{\sqrt {a}} \]

[In]

int((a*x + 1)^(1/2)/x^(1/2),x)

[Out]

x^(1/2)*(a*x + 1)^(1/2) + (2*atanh((a^(1/2)*x^(1/2))/((a*x + 1)^(1/2) - 1)))/a^(1/2)